Summary
- Stellar variability can contain information about potential exoplanets.
- But signal is buried amongst other effects: granulation, supergranulation, oscillations, activity.
- Looking to detect QPOs in photometric and/or radial velocity observations.
- Suggest that time-domain analysis with GPs is better than frequency domain with PSDs.
- Simulation study demonstrating this claim.
Methodology
- Simulate light curve with known PSD parameters.
- Estimate PSD:
- Discrete Fourier Transform (FFT)
- Generalised Lomb-Scargle (GLS)
- Fit model to PSD (using MCMC after binning).
- Fit GP to simulated light curve.
- Compare with known parameters.
PSD Models
Parameters: amplitude \(S_0\), frequency \(\nu_0\), quality (damping) factor \(Q\).
\[P_\textrm{SHO}(\nu) = \frac{2 S_0 \nu_0^4}{(\nu^2 - \nu_0^2)^2 + \frac{\nu^2 \nu_0^2}{Q^2}}\]
Aperiodic
\[P_\textrm{SHO}(\nu)\rvert_{Q = \frac{1}{\sqrt{2}}} = \frac{2 S_0}{1 + \left(\frac{\nu}{\nu_0}\right)^4}\]
c.f., background granulation noise.
Scenarios
- Single aperiodic component.
- Single periodic component.
- Two harmonic (“rotational”) components.
- Periodic + aperiodic.
- Two aperiodic components.
Evenly sampled vs. Unevenly sampled.
Data
- Simulated light curves
- 288 days with 100 points per day
- Draw samples from
celerite GP
- periodic + aperiodic terms
- white noise term
- Subsampled according to timestamps from HARPS-N solar telescope observations between 2015 and 2018.
Windowing Function
Irregularly sampled \(g(t)\) as the convolution of continuous function \(f(t)\) and window function \(w(t)\).
\[g(t) = f(t) \otimes w(t)\]
where \(w(t)\) is approximated by a sum of delta \(\delta()\) functions.
In frequency domain:
\[G(\nu) = F(\nu) \times W(\nu)\]
Windowing
Results: Single aperiodic
Results: Single aperiodic
Results: Single periodic
Results: Two harmonics
Results: Aperiodic + periodic
Results: Two aperiodic
Results
Conclusions
- GP regression in time domain gives better results than PSD modelling in frequency domain.
- Irregular sampling is the nemesis of PSD estimators.
- Revisit recent studies that used PSD estimation for analysis of radial velocities for exoplanet searches.
- Expect similar with PSDs for QPOs in X-ray binaries, AGNs.
- There is a computation cost to GPs
Comments